MaBloWriMo 9: omega and its ilk

So far, we have defined a sequence of numbers $s_n$ , and showed that

$s_n = \omega^{2^n} + \overline{\omega}^{2^n}$

where $\omega = 2 + \sqrt 3$ and $\overline{\omega} = 2 - \sqrt 3$ . This is a big step: the $s_n$ are defined recursively (that is, each $s_n$ is defined in terms of the previous $s_{n-1}$ ), but $\omega$ and $\overline{\omega}$ give us a direct arithmetic expression for $s_n$ , which can make proving things about the $s_n$ easier. So our next goal will be to better understand $\omega$ .

As a warmup, consider that $\omega$ and $\overline{\omega}$ are both of the form $a + b\sqrt 3$ , where $a$ and $b$ are integers. Consider the set of all such numbers, call it $S$ . So, for example, $S$ contains things like $4 - 19\sqrt{3}$ and $0 + \sqrt{3}$ and $99 + 999\sqrt{3}$ and $2 + 0\sqrt{3}$ … and so on. (Mathematicians would call this set $\mathbb{Z}[\sqrt{3}]$ , but it doesn’t really matter what we call it.) First of all, note that if we take any two numbers in $S$ and add them, we get another number in $S$ :

$(a + b\sqrt{3}) + (c + d\sqrt{3}) = (a + c) + (b + d)\sqrt{3}$

For example, $(2 + 3\sqrt{3}) + (4 - 4\sqrt{3}) = 6 - \sqrt{3}$ . A shorter way to say this is that $S$ is “closed under addition”. The additive identity, $0$ , is also a member of $S$ , since $0 = 0 + 0\sqrt{3}$ .

We can show that $S$ is closed under multiplication, too:

$(a + b\sqrt{3})(c + d\sqrt{3}) = ac + bc\sqrt{3} + ad\sqrt{3} + 3bd = (ac + 3bd) + (bc + ad)\sqrt{3}$

And the multiplicative identity $1$ is likewise in $S$ , since $1 = 1 + 0 \sqrt{3}$ .

So $S$ is a sort of “system of numbers” in a similar sense to the integers, or rational numbers, or especially the complex numbers $a + bi$ . At this point we could start talking about rings and fields and stuff, but we don’t actually need anything that complicated. We just need to talk a little bit about groups. I’ve been wanting to write about groups on this blog for a while, and this is a great excuse! So, tomorrow, we’ll start to learn about groups and develop some of the ideas we will need to understand $\omega$ .

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.

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